Random Reaction Networks Research Articles

Universal Transitions between Growth and Dormancy via Intermediate Complex Formation.

A simple cell model consisting of a catalytic reaction network with intermediate complex formation is numerically studied. As nutrients are depleted, the transition from the exponential growth phase to the growth-arrested dormant phase occurs along with hysteresis and a lag time for growth recovery. This transition is caused by the accumulation of intermediate complexes, leading to the jamming of reactions and the diversification of components. These properties are generic in random reaction networks, as supported by dynamical systems analyses of corresponding mean-field models.

Prevalence of multistability and nonstationarity in driven chemical networks.

External flows of energy, entropy, and matter can cause sudden transitions in the stability of biological and industrial systems, fundamentally altering their dynamical function. How might we control and design these transitions in chemical reaction networks? Here, we analyze transitions giving rise to complex behavior in random reaction networks subject to external driving forces. In the absence of driving, we characterize the uniqueness of the steady state and identify the percolation of a giant connected component in these networks as the number of reactions increases. When subject to chemical driving (influx and outflux of chemical species), the steady state can undergo bifurcations, leading to multistability or oscillatory dynamics. By quantifying the prevalence of these bifurcations, we show how chemical driving and network sparsity tend to promote the emergence of these complex dynamics and increased rates of entropy production. We show that catalysis also plays an important role in the emergence of complexity, strongly correlating with the prevalence of bifurcations. Our results suggest that coupling a minimal number of chemical signatures with external driving can lead to features present in biochemical processes and abiogenesis.

Deficiency zero for random reaction networks under a stochastic block model framework

Deficiency zero is an important network structure and has been the focus of many celebrated results within reaction network theory. In our previous paper Prevalence of deficiency zero reaction networks in an Erdős-Rényi framework, we provided a framework to quantify the prevalence of deficiency zero among randomly generated reaction networks. Specifically, given a randomly generated binary reaction network with n species, with an edge between two arbitrary vertices occurring independently with probability \(p_n\), we established the threshold function \(r(n)=\frac{1}{n^3}\) such that the probability of the random network being deficiency zero converges to 1 if \(\frac{p_n}{r(n)}\rightarrow 0\) and converges to 0 if \(\frac{p_n}{r(n)}\rightarrow \infty \), as \(n \rightarrow \infty \). With the base Erdős-Rényi framework as a starting point, the current paper provides a significantly more flexible framework by weighting the edge probabilities via control parameters \(\alpha _{i,j}\), with \(i,j\in \{0,1,2\}\) enumerating the types of possible vertices (zeroth, first, or second order). The control parameters can be chosen to generate random reaction networks with a specific underlying structure, such as “closed” networks with very few inflow and outflow reactions, or “open” networks with abundant inflow and outflow. Under this new framework, for each choice of control parameters \(\{\alpha _{i,j}\}\), we establish a threshold function \(r(n,\{\alpha _{i,j}\})\) such that the probability of the random network being deficiency zero converges to 1 if \(\frac{p_n}{r(n,\{\alpha _{i,j}\})}\rightarrow 0\) and converges to 0 if \(\frac{p_n}{r(n,\{\alpha _{i,j}\})}\rightarrow \infty \).

Non-normality and non-monotonic dynamics in complex reaction networks

Complex chemical reaction networks, which underlie many industrial and biological processes, often exhibit non-monotonic changes in chemical species concentrations, typically described using nonlinear models. Such non-monotonic dynamics are in principle possible even in linear models if the matrices defining the models are non-normal, as characterized by a necessarily non-orthogonal set of eigenvectors. However, the extent to which non-normality is responsible for non-monotonic behavior remains an open question. Here, using a master equation to model the reaction dynamics, we derive a general condition for observing non-monotonic dynamics of individual species, establishing that non-normality promotes non-monotonicity but is not a requirement for it. In contrast, we show that non-normality is a requirement for non-monotonic dynamics to be observed in the R\'enyi entropy. Using hydrogen combustion as an example application, we demonstrate that non-monotonic dynamics under experimental conditions are supported by a linear chain of connected components, in contrast with the dominance of a single giant component observed in typical random reaction networks. The exact linearity of the master equation enables development of rigorous theory and simulations for dynamical networks of unprecedented size (approaching $10^5$ dynamical variables, even for a network of only 20 reactions and involving less than 100 atoms). Our conclusions are expected to hold for other combustion processes, and the general theory we develop is applicable to all chemical reaction networks, including biological ones.

Molecular Codes in Biological and Chemical Reaction Networks

Shannon’s theory of communication has been very successfully applied for the analysis of biological information. However, the theory neglects semantic and pragmatic aspects and thus cannot directly be applied to distinguish between (bio-) chemical systems able to process “meaningful” information from those that do not. Here, we present a formal method to assess a system’s semantic capacity by analyzing a reaction network’s capability to implement molecular codes. We analyzed models of chemical systems (martian atmosphere chemistry and various combustion chemistries), biochemical systems (gene expression, gene translation, and phosphorylation signaling cascades), an artificial chemistry, and random reaction networks. Our study suggests that different chemical systems posses different semantic capacities. No semantic capacity was found in the model of the martian atmosphere chemistry, the studied combustion chemistries, and highly connected random networks, i.e. with these chemistries molecular codes cannot be implemented. High semantic capacity was found in the studied biochemical systems and in random reaction networks where the number of second order reactions is twice the number of species. We conclude that our approach can be applied to evaluate the information processing capabilities of a chemical system and may thus be a useful tool to understand the origin and evolution of meaningful information, e.g. in the context of the origin of life.

Dynamical stability of autocatalytic sets

Theoretical investigations of autocatalytic sets rendere d the occurrence of self-sustaining sets of molecules to be a generic property of random reaction networks. This stands in some contrast to the experimental difficulty to actually find such systems. In this work, we argue that the usual approach, which is based on the study of static properties of reaction graphs has to be complemented with a dynamic perspective in order to avoid overestimation of the probability of getti ng autocatalytic sets. Especially under the, from the experimental point of view, important flow reactor conditions, it is no t sufficient just to have a pathway generating a given type of molecules. The respective process has also to happen with a sufficient rate in order to compensate the outflow. Reaction rates are therefore of crucial importance. Furthermore, pr ocesses such as cleavage are on one hand advantageous for the system, because they enhance the molecular variability and therefore the potential for catalysis. On the other hand, cl eavage may also act in an inhibiting manner by the destruction of vital components: therefore, an optimal balance between ligation and cleavage has to be found. If energy is included as a limiting resource, the concentration profiles of the compo nents of autocatalytic sets are altered in a manner that rend ers a certain range for the energy supply rate as optimal for the realization of robust autocatalytic sets. The results presented are based on a theoretical model and obtained by numerical integration of systems of ODE. This limits the number of involved molecular species which implies that the quantitative findings of this work may have no direct relevance for experimental situations, whereas the qualit ative insights in the dynamics of the systems under consideration may generalize to systems of truly combinatorial size.